Abstract
If properties are to play a useful role in semantics, it is hard to avoid assuming the naive theory of properties: for any predicate Θ(x), there is a property such that an object o has it if and only if Θ(o). Yet this appears to lead to various paradoxes. I show that no paradoxes arise as long as the logic is weakened appropriately; the main difficulty is finding a semantics that can handle a conditional obeying reasonable laws without engendering paradox. I employ a semantics which is infinite-valued, with the values only partially ordered. Can the solution be adapted to naive set theory? Probably not, but limiting naive comprehension in set theory is perfectly satisfactory, whereas this is not so in a property theory used for semantics.
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